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Zelditch's proof of the Tian–Yau–Zelditch Theorem makes use of the Boutet de Monvel–Sjöstrand results on the asymptotic properties of Szegö projectors for strictly pseudoconvex domains. However, as we will show below, the theorem is also closely related to another theorem of Boutet de Monvel's, namely his wave trace formula for Toeplitz operators. Finally, we will derive, for the pseudoconvex manifolds considered by Zelditch in his proof of the Tian–Yau–Zelditch Theorem, an analogue of another result of Boutet de Monvel's, the extendability theorem of Berndtsson for holomorphic functions on Grauert tubes.

We give a new proof of former results by $\text{G}$. Zampieri and the author on extension of holomorphic functions from one side $\Omega$ of a real hypersurface $M$ of ${{\mathbb{C}}^{n}}$ in the presence of an analytic disc tangent to $M$, attached to $\bar{\Omega }$ but not to $M$. Our method enables us to weaken the regularity assumptions both for the hypersurface and the disc.

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The Tian–Yau–Zelditch Theorem and Toeplitz Operators

Extension of Holomorphic Functions From One Side of a Hypersurface

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2 results

The Tian–Yau–Zelditch Theorem and Toeplitz Operators

Extension of Holomorphic Functions From One Side of a Hypersurface